OVER the past century, scientists have faced many challenges to their
cherished beliefs and emerged stronger and wiser. They have learned to live with
the uncertainty principle. And they have accepted their inability to predict
such non-linear phenomena as the weather. Another dearly-held belief, however,
is proving rather more resilient.
Researchers are still reluctant to acknowledge the crucial role that
subjectivity plays in science. Informally, of course, many will agree that they
often act on hunches or beliefs, and that the process of turning raw figures
into insights is not wholly objective. But ask them to factor subjectivity into
their equations, and they take fright. It suddenly seems like an attempt to
reduce science to the relativisms of literary criticism.
Nowhere is this reluctance clearer than in the controversy surrounding the
use of Bayes鈥檚 theorem. Based on work done nearly 250 years ago by the Reverend
Thomas Bayes, an English cleric-cum-mathematician, the theorem offers a powerful
way of assessing the significance of new findings. It would be a boon if it were
used more widely (快猫短视频, 22 November 1997, p 36).
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The problem with Bayes鈥檚 theorem is it forces scientists to say how plausible
they think the hypothesis they are testing is鈥攁nd put a hard number on
that belief, even if it鈥檚 based on nothing more than a hunch. So while there has
been lots of talk about the usefulness of Bayes鈥檚 theorem, and growing numbers
of papers explaining how the method works, few studies have actually used the
theorem to settle big statistical controversies such as whether power lines can
cause cancer.
This reluctance to accept a powerful way of making sense of new findings is
bad enough. But the situation is more serious than that, as is made clear in
what I think is one of the most important neglected papers of the 20th century.
In the mid-1940s, an American physicist called Richard Cox from Johns Hopkins
University, Maryland, proved that if you want to make sense of scientific
evidence, putting hard numbers on beliefs and hunches is not only
handy鈥攊t鈥檚 the only way of doing it logically and consistently.
Cox asked himself whether it was possible to deduce laws for scientific
reasoning. He knew that rules for dealing with absolute truth and falsity had
been worked out a century earlier by the British mathematician George Boole. But
Cox wanted to know whether there were rules for reasoning about statements that
lie somewhere between these two extremes.
He began by sketching out some of the demands he could make of such
rules鈥攆or example, that the rules reflect the notion that the more we
believe a proposition to be true, the less we believe it to be false. Then Cox
used symbolic logic and algebra to show scientific reasoning does indeed obey
certain rules. And those rules turn out to be identical to the rules of
probability theory.
In retrospect, that may not seem so surprising. But within Cox鈥檚 proof,
published in the American Journal of Physicsin 1946, there lurks a
serpent for anyone who thinks that scientific reasoning is utterly
objective.
The very essence of science is the assessment of theories in the light of
evidence: that is, deciding how we should change how much we believe in a
hypothesis, given the information we鈥檝e collected. Cox showed that the way to do
this logically and consistently is via the appropriate rule from probability
theory. And that rule turns out to be Bayes鈥檚 theorem.
At root, then, the theorem is merely a means of updating one鈥檚 beliefs in a
hypothesis in the light of new evidence. That clearly demands some beliefs to
update in the first place. But what if you鈥檙e the first person to investigate
some hypothesis? The answer, according to Bayes鈥檚 theorem, is simple: just use
your intuition, experience and judgment.
Does that mean that science is no better than an intellectual free-for-all,
ruled by opinion rather than hard fact? Not at all. Suppose you鈥檙e trying to
find out whether professional comedians are more likely to be depressives.
Chances are you don鈥檛 have any solid evidence, just a hunch based on the lives
of the likes of Lenny Bruce, Tony Hancock, and Phil Silvers鈥攁ll great
comedians, and all great depressives.
Scientifically, the next step is clear: get some hard data鈥攂y giving
psychiatric tests to hundreds of comedians, for example. Bayes鈥檚 theorem will
then show how you should update your original beliefs in the light of some solid
statistics. And as you accumulate more information, Bayes鈥檚 theorem shows that
your original thoughts 鈥攆laky or well-founded, right or wrong鈥攂ecome
progressively less important. It鈥檚 the hard facts that come to dominate your
conclusion鈥攈ardly a 鈥渃ontroversial鈥� way of dealing with evidence.
And that is what makes Cox鈥檚 result so crucial. By revealing the central
importance of Bayes鈥檚 theorem in scientific reasoning, Cox showed not only that
the scientific process contains an ineluctable amount of subjectivity at the
outset, but that it gives way to objectivity as the information accumulates. In
other words, scientific objectivity is 鈥渆mergent鈥�.
Conventional statistical tools ignore this completely. We鈥檝e all come across
some new research claiming to have found a wonder drug for a killer disease,
say, or a link between cancer and some environmental pollutant鈥攁nd rolled
our eyes in scepticism. Today鈥檚 scientific journals are full of such research.
Their more-or-less implausible findings have only one claim to reliability: that
they鈥檙e 鈥渟tatistically significant鈥�. This is one reason why we have seemingly
endless debates over whether living near pylons causes leukaemia.
Bayes鈥檚 theorem would compel those who want to keep the debate alive to show
that their claims can overturn the weight of statistical evidence against any
link鈥攐r not, as the case may be. It demands that new findings be assessed
in the light of previous experience, thus giving a much better handle on their
plausibility.
There is an alternative, of course. We could continue with the pretence that
Bayes鈥檚 theorem is just a technical result of interest only to statistics buffs,
and that subjectivity and equations don鈥檛 mix. But as Cox showed 50 years ago,
we can hardly then complain if we end up with 鈥渟tatistically significant鈥�
findings which seem neither logical nor consistent.
We鈥檝e learnt to live with the uncertainty principle and unpredictability.
It鈥檚 time we learned to live with Bayes鈥檚 theorem too.